Предельные теоремы Сегё

Предельные теоремы Сегё — группа результатов, описывающих асимптотическое поведение детерминантов больших теплицевых матриц^[1], впервые установленная Габором Сегё.

В рамках предельных теорем рассматривается функция $\varphi \colon T\to \mathbb {C}$ , заданная на единичной окружности комплексной плоскости, и тёплицева матрица $T_{n}(\varphi )$ размера $n\times n$ , определённая как:

T_{n}(\phi )_{k,l}={\widehat {\phi }}(k-l),\quad 0\leqslant k,l\leqslant n-1

где

{\widehat {\phi }}(k)={\frac {1}{2\pi }}\int _{0}^{2\pi }\phi (e^{i\theta })e^{-ik\theta }\,d\theta

являются коэффициентами Фурье функции $\varphi$ .

Первая теорема Сегё^[1]^[2] утверждает, что при $\varphi >0$ и $\varphi \in L_{1}(T)$ справедливо:

\lim _{n\to \infty }{\frac {\det T_{n}(\phi )}{\det T_{n-1}(\phi )}}=\exp \left\{{\frac {1}{2\pi }}\int _{0}^{2\pi }\log \phi (e^{i\theta })\,d\theta \right\}.

Правая часть является геометрическим средним функции $\varphi$ (которое определено в силу соотношения между геометрическим и арифметическим средними; обозначается через $G(\varphi )$ ).

Вторая теорема Сегё^[1]^[3] (строгая теорема Сегё) утверждает, что если дополнительно потребовать, чтобы производная $\varphi$ была гёльдеровой функцией порядка $\alpha >0$ , то справедливо:

\lim _{n\to \infty }{\frac {\det T_{n}(\phi )}{G^{n}(\phi )}}=\exp \left\{\sum _{k=1}^{\infty }k\left|{\widehat {(\log \phi )}}(k)\right|^{2}\right\}.

Примечания

↑ ¹ ² ³ Böttcher, Albrecht; Silbermann, Bernd. Toeplitz determinants // Analysis of Toeplitz operators (неопр.). — Berlin: Springer-Verlag, 1990. — С. 525. — ISBN 3-540-52147-X.
↑ Szegő, G. Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion (нем.) // Math. Ann. : magazin. — 1915. — Bd. 76, Nr. 4. — S. 490—503. — doi:10.1007/BF01458220.
↑ Szegő, G. On certain Hermitian forms associated with the Fourier series of a positive function (англ.) // Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] : journal. — 1952. — P. 228—238.

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